Feedback nonlinear equalization of modulated data signals

ABSTRACT

A receiver for a quadrature amplitude modulated data signal impaired by linear and nonlinear distortion, phase jitter and additive noise includes circuitry which compensates for these impairments. In particular, the receiver includes a processor (FIG. 1 44; FIG. 2, 44&#39;) which subtracts a feedback nonlinear signal (FIG. 1, D(n); FIG. 2, D&#39;(n)) from each sample of the received signal, either prior or subsequent to demodulation, providing compensation for nonlinear intersymbol interference. The feedback nonlinear signal subtracted from each sample is comprised of a weighted sum of products of individual data decisions and/or the complex conjugates of data decisions, each such product, in turn, being multiplied by a predetermined harmonic of the carrier frequency. In an illustrative embodiment, compensation for second- and third-order intersymbol interference is provided by including two- and three-multiplicand weighted products in the feedback nonlinear signal. Weighting coefficients for each product are adaptively updated in a decision-directed manner.

CROSS-REFERENCE TO RELATED APPLICATION

My U.S. patent application entitled "Feedforward Nonlinear Equalizationof Modulated Data Signals", Ser. No. 931,026, was filed in the U.S.Patent and Trademark Office concurrently herewith.

BACKGROUND OF THE INVENTION

My invention relates to the correction of the distorting effects oflimited bandwidth transmission media on modulated data signals.

The principal impediment to accurate reception of high-speed datasignals transmitted over limited bandwidth, e.g., switched voicebandtelephone, transmission channels is that form of distortion known asintersymbol interference. This phenomenon is a manifestation of the factthat a pulse passing through a band-limited channel expands in the timedomain. As a result, each sample of the received signal is not simplyderived from a single transmitted data symbol but, rather, somecombination of symbols. Other impairments include phase jitter andadditive noise.

Linear intersymbol interference, in particular, is manifested in thateach sample of the received signal contains a linear combination of atransmitted symbol--which the sample nominally represents--with symbolswhich precede and succeed it in the data stream. Among known techniqueswhich compensate for the distorting effects of linear intersymbolinterference in both baseband and passband, e.g., quadrature amplitudemodulated (QAM), signals are linear feedforward equalization and lineardecision feedback equalization. In accordance with the former technique,each sample of the received signal is weighted with a linear sum of pastand future samples prior to a decision being made as to the value of thetransmitted symbol. In accordance with the latter technique, a weightedlinear sum of past decisions is combined with each sample, again priorto a decision being made as to the value of the transmitted symbol. See,for example, my U.S. Pat. No. 3,974,449 issued Aug. 10, 1976.

Nonlinear intersymbol interference is manifested in that each sample ofthe received signal includes a linear combination of products of thecurrent, past and future modulated data symbols, and/or (in the case ofQAM, for example) the complex conjugates of such data symbols. Intransmission systems that employ linear modulation, such as QAM, theeffect is to reduce the margin against noise. Indeed, for data ratesabove 4800 bps, nonlinear distortion is the dominant impairment on manyvoiceband channels. At least one arrangement is known which compensatesfor nonlinear intersymbol interference in baseband data signals. See,e.g., U.S. Pat. No. 3,600,681 issued Aug. 17, 1971 to T. Arbuckle.However, the known arrangements will not, in general, effectivelycompensate for nonlinear intersymbol interference in modulated datasignals.

SUMMARY OF THE INVENTION

An object of my invention is to provide a method and arrangement whichcompensates for nonlinear intersymbol interference in modulated datasignals.

A more particular object of my invention is to provide a method andarrangement which compensates for nonlinear intersymbol interference inmodulated data signals in which both the carrier phase and amplitude aremodulated, i.e., information-bearing.

A still more particular object of my invention is to provide a methodand arrangement which compensates for nonlinear intersymbol interferencein quadrature amplitude modulated data signals.

In accordance with the invention, the above and other objects areachieved by combining with each sample of the received signal anassociated feedback nonlinear signal. The feedback nonlinear signalincludes a weighted sum of products of (a) data decisions made onindividual demodulated samples, and (b) complex conjugates of such datadecisions. Each data decision/complex conjugate product, in turn,modulates a predetermined harmonic of the carrier frequency.

Each data decision/complex conjugate product has a predetermined numberof multiplicands, i.e., data decisions and/or complex conjugates, eachof which bears a predetermined temporal relationship to the associatedsample. In general, inclusion in the feedback nonlinear signal ofproducts having a total of m data decisions and complex conjugatesprovides compensation for m^(th) order intersymbol interference.

The coefficients used in weighting the various products areillustratively updated in an adaptive, decision-directed manner.

BRIEF DESCRIPTION OF THE DRAWING

In the drawing,

FIGS. 1 and 2 are block diagrams of first and second illustrativeembodiments, respectively, of receivers for modulated data signalsincluding equalization circuitry which combines with each sample anonlinear feedback signal in accordance with the invention;

FIG. 3 is a block diagram of an illustrative feedback nonlinear signalprocessor for use in the receivers of FIGS. 1 and 2;

FIG. 4 is an illustrative coefficient store and multiplier unit for usein the processor of FIG. 3.

DETAILED DESCRIPTION

The receiver of FIG. 1 is illustratively employed in a high-speedtelephone-voiceband data transmission system using quadrature amplitudemodulation (QAM). The sampling interval is T seconds, the signaling ratebeing 1/T symbols per second. QAM entails both phase and amplitudemodulation of a carrier, i.e., both the carrier phase and amplitude areinformation-bearing. As a result, QAM signals are referred to as"complex" signals and can be represented for notational convenience ascomplex numbers. This notational convention is followed herein so thatall of the signal reference letters used in the following descriptionshould be understood to represent complex signals.

The receiver FIG. 1 is of the same general type as that disclosed in myU.S. Pat. No. 3,974,449 issued Aug. 10, 1976, which is herebyincorporated by reference. Thus, as in my earlier patent, a sample R(n)of a received QAM data signal is provided on input lead 10, the index nindicating that R(n) is the sample of the QAM signal at time nT. SampleR(n) is applied to a feedforward processor 11. After some delay, thelatter generates an equalized version of sample R(n), feedforward signalG(n), on lead 35, thereby providing at least some compensation forintersymbol interference in sample R(n) as well as for some of theadditive noise present therein. Feedforward processor 11 may include afeedforward linear processor of conventional design for generating aspart of signal G(n) a linear combination of past, present and futurereceived samples, providing compensation for linear intersymbolinterference. If desired, feedforward processor 11 may also include afeedforward nonlinear processor of the type disclosed in myabove-referenced copending U.S. patent application. Such a feedforwardnonlinear processor generates as part of signal G(n) a linearcombination of products of samples on lead 10 and their complexconjugates, thereby providing at least some compensation for nonlinearintersymbol interference.

Signal G(n) is extended to a demodulator 12 over lead 35. Demodulator 12produces a demodulated baseband data signal Z(n) which is applied to oneinput of a subtractor 14.

The receiver of FIG. 1 further includes data recovery circuit, orquantizer, 17. This unit quantizes the output signal of subtractor14--data recovery input signal Y(n)--to form a decision A(n) as to thevalue of the original modulating data symbol represented by, and to berecovered from, sample R(n). (Quantization of complex signals amounts topartitioning the complex plane into decision regions surrounding theideal received points.) Decision A(n) passes on to data sink 25.Decision A(n) is also applied to feedback linear processor 15 over lead31.

Feedback linear processor 15 operates in conventional fashion togenerate a feedback linear signal C(n) on lead 16. Signal C(n) iscomprised of a linear combination of decisions made by data recoverycircuit 17 prior to decision A(n). Processor 15 includes a delay line15a to facilitate the formation of signal C(n). Signal C(n) is combinedwith--illustratively subtracted from--signal Z(n) in a manner describedbelow to form the abovementioned signal Y(n), thereby removing at leasta portion of the linear intersymbol interference and additive noise notcompensated for upstream in feedforward processor 11.

In accordance with the present invention, a feedback nonlinear signalD(n) is provided on lead 45 by a feedback nonlinear processor 44. SignalD(n) is also subtracted from signal Z(n), thereby removing at least aportion of the nonlinear intersymbol interference and additive noise notcompensated for upstream in processor 11. As described in detail below,signal D(n) is comprised of a weighted sum of products of decisions madeby data recovery circuit 17 and the complex conjugates of such datadecisions, each product, in turn, modulating a predetermined harmonic ofthe carrier frequency. Signals C(n) and D(n) are illustrativelysubtracted from signal Z(n) by first adding them together in an adder 46generate a composite feedback signal V(n). The latter is then subtractedfrom a signal Z(n) in subtractor 14 to generate signal Y(n).

So-called weighting, or tap, coefficients for forming the aforementionedcombinations of (a) received signal samples and products of signalsamples and complex conjugates thereof in feedforward processor 11, (b)data decisions in feedback linear processor 15, and (c) products of datadecisions and complex conjugates thereof in feedback nonlinear processor44, are automatically adjusted in an adaptive, decision-directed manner.This automatic adjustment of tap coefficients is implemented bycircuitry including error computer 20. This unit provides on lead 22 thecomplex conjugate, E(n), of an estimated error signal E(n), the latterrepresenting the difference between signal Y(n) and decision A(n). Thetap coefficients used in processors 15 and 44 are adjusted in responseto signal E(n) in such a way as to minimize the average squaredmagnitude of that signal. The tap coefficients used in processor 11 areadjusted in response to the complex conjugate of a modulated version ofthe estimated error signal--signal J(n)--again in such a way as tominimize the average squared magnitude of that signal. Signal J(n) isprovided to processor 11 by remodulator 28 on lead 27.

Phase jitter and frequency offset in modulated data sample R(n) canhinder accurate data recovery. In order to compensate for theseimpairments, demodulator 12 and remodulator 28 perform their functionsusing complex exponential signals of the forme^(-j)[2πf.sbsp.c^(nT+)θ(n)], which are generated by carrier recoverycircuit 24. The phase angle θ(n) is an estimate of the carrier phaseθ(n) of sample R(n). The estimated phase θ(n+1) during the (n+1)^(st)sampling period is updated in accordance with

    θ(n+1)=θ(n)-α(n)Im[J(n)G(n)]             (1)

Carrier recovery circuit 24 receives signals J(n) and G(n) for purposesof computing θ(n+1) in accordance with Eq. (1).

The factor α(n) in Eq. (1) may simply be a constant stored within thecarrier recovery circuit 24. Alternatively, factor α(n) may be afunction of current signal values so that updating of θ(n) is carriedout in response only to the phase angle error [θ(n)-θ(n)] and not inresponse to errors due to imperfect equalization and random amplitudemodulation by data symbols in processors 11, 15 and 44. In deriving anexpression for such an α(n), perfect equalization is postulated byassuming that the only discrepancy between Y(n) and A(n) is in the phaseerror, [θ(n)-θ(n)]. Therefore,

    Z(n)≈[A(n)+V(n)]e.sup.[θ(n)-θ(n)].     (2)

Moreover, E(n)=Z(n)-V(n)-A(n) and it can be shown that J*(n)G(n) ismathematically equivalent to E*(n)Z(n). Therefore, Eq. (2) can besubstituted into Eq. (1) to yield

    θ(n+1)=θ(n)-α(n)|A(n)+V(n)|.sup.2 sin (θ(n)-θ(n)).

Therefore, a suitable choice for α(n) is seen to be

    α(n)=α/|A(n)+V(n)|.sup.2,    (3)

where α is a small constant, because then

    θ(n+1)=θ(n)-α sin (θ(n)-θ(n)).

Thus, as desired, the updating of θ(n) is based only on the phase error,providing a smoother, more direct acquisition of carrier phase. Carrierrecovery circuit 24 illustratively receives decision A(n) and signalV(n) for purposes of generating α(n) in accordance with Eq. (3).

FIG. 2 illustrates an alternate embodiment of a receiver which includesa feedback nonlinear processor in accordance with the invention. Here,the output of the feedback nonlinear processor 44'--feedback nonlinearsignal D'(n)--is subtracted from signal G(n) in a subtractor 46' to forma combined modulated signal F(n) rather than being subtracted fromsignal Z(n) via adder 46 as in FIG. 1. Signal D'(n) differs from signalD(n) in that different harmonics multiply each data decision/complexconjugate product, as described below. In addition, signal V(n) isgenerated for purposes of carrier recovery, as previously described, bypassing signal D'(n) through a demodulator 12' and adding the outputthereof to signal C(n) in an adder 13. The embodiments of FIGS. 1 and 2are otherwise similar, however, with corresponding elements having thesame reference numeral in each FIG. Thus the preceding discussionrelating to the structure and operation of the receiver of FIG. 1 isgenerally applicable to the receiver of FIG. 2.

With the exception of processor 44 (FIG. 1), processor 44' (FIG. 2), andthe feedforward nonlinear circuitry in processor 11, if any, thespecific circuitry comprising the various components of the receivers ofFIGS. 1 and 2, as well as their functional and timinginterrelationships, are all well known in the art and need not bediscussed in further detail. See, for example, my above-cited U.S.Patent for a description of the receiver generally and my above-citedcopending U.S. Patent application for a description of the feedforwardnonlinear circuitry. The remainder of this detailed description, then,is principally directed to (a) characterization of feedback nonlinearsignals D(n) and D'(n) and (b) description of illustrative circuitry,shown in FIGS. 3 and 4, for generating them.

In accordance with the present invention, the feedback nonlinear signalassociated and combined with each sample of the received signal includesa weighted sum of products of individual ones of the data decisions madeby data recovery circuit 17 and complex conjugates of such datadecisions. Each data decision/complex conjugate product--hereinafterreferred to for generality as "decision/conjugate product" even though aparticular product may not have any complex conjugates--in turn,modulates a predetermined harmonic of the carrier frequency, therebyproviding a plurality of modulated weighted products. Eachdecision/conjugate product has a predetermined number of multiplicands,each of which bears a predetermined temporal relationship to theassociated sample. In general, inclusion in the feedback nonlinearsignal of products having a total of m data decisions and complexconjugates provides compensation for m^(th) order intersymbolinterference.

In the case where the feedback nonlinear signal is added to theassociated sample after demodulation, as in FIG. 1, the above-mentionedharmonic is of the general form

    e.sup.j[2πf.sbsp.c.sup.nT(x-y-1)+Φ.sbsp.x,y.sup.],  4(a)

where

f_(c) =carrier frequency

n=sample time index

x=number of data decisions in the decision/conjugate product which theharmonic multiplies

y=number of complex conjugates in the decision/conjugate product whichthe harmonic multiplies

Φ_(x),y =a selected phase angle (discussed below).

In the case where the feedback nonlinear signal is added to theassociated sample prior to demodulation, as in FIG. 2, the harmonic isof the general form

    e.sup.j[2πf.sbsp.c.sup.nT(x-y)+Φ.sbsp.x,y.sup.].    4(b)

The phase angle Φ_(x),y in Eqs. 4(a) and 4(b) may be, for example, aterm which corrects for distortion due to phase jitter and/or frequencyoffset in the nonlinear terms of each received sample, i.e., theproducts of modulated data symbols which constitute the nonlinearintersymbol interference to be removed from the sample. If at least someof this distortion was introduced in the transmission system prior tothe nonlinearity which caused the nonlinear intersymbol interference,different values of Φ_(x),y may have to be selected for each combinationof values of the parameters x and y.

In the embodiments disclosed herein, however, it is presumed that anyphase jitter or frequency offset was introduced after the nonlinearity.As a result, all the Φ_(x),y 's may have the same value. In particular,any phase jitter and/or frequency offset in the nonlinear terms iscompensated for in the embodiment of FIG. 1 by virtue of the fact thatthose terms are removed from each sample after a demodulation whichmultiplies the sample by e^(-j)[2πf.sbsp.c^(nT+)θ(n)]. Thus each Φ_(x),ycan be set to an arbitrary constant, illustratively zero. In theembodiment of FIG. 2, on the other hand, the nonlinear terms are removedfrom the sample being processed prior to demodulation. In order tocompensate in this embodiment for phase jitter and/or frequency offsetin the nonlinear terms, each Φ_(x),y is illustratively set to θ(n).

Attention is now particularly directed to the embodiment of FIG. 1.Processor 44 thereof illustratively provides compensation for second-and third-order nonlinear intersymbol interference. Thus, compositefeedback signal V(n) can be expressed as follows:

    V(n)=[C(n)+D(n)]                                           (5)

where ##EQU1## C(n), the prior art feedback linear signal, is comprisedof a linear combination of data decisions A(n-i), each sample beingweighted by the complex conjugate of complex weighting, or tap,coefficient ##EQU2## The index i typically spans a range of positivevalues ≧1 so that C(n) includes a sufficient number of data decisionsmade in data recovery circuit 17 prior to decision A(n) to yieldeffective equalization. As is conventional, the values of coefficients##EQU3## are adjusted adaptively in a decision-directed manner inprocessor 15 illustratively using a gradient adaptation algorithm,yielding the updating relationship ##EQU4## with γ₀ (n) being a selectedfactor which may be updated each sampling period or which may moresimply be a constant.

The first two terms of feedback nonlinear signal D(n), defined forconvenience as D₁ (n) and D₂ (n), provide compensation for second-orderintersymbol interference. (In a given application, it may be desired toinclude only one of these terms in signal D(n).) Each of the terms D₁(n) and D₂ (n) is comprised of a weighted sum of two-multiplicandproducts which are modulated, in turn, by a harmonic of the carrierfrequency defined by Eq. 4a. Each multiplicand of each product isderived from a selected data decision. That is, each multiplicand iseither a data decision or the complex conjugate of a data decision.

In particular, the two multiplicands of each decision/conjugate productof term D₁ (n) are a selected two data decisions A(n-j₁) and A(n-j₂)weighted by the complex conjugate of an associated weightingcoefficient. ##EQU5## and modulated by e^(j2)πf.sbsp.c^(nT) (all theΦ_(x),y 's being zero in this embodiment, as discussed above). Indexpairs (j₁,j₂) are predetermined and are selected keeping in mind thatthe nonlinear (and, indeed, linear) intersymbol interference in a datadecision can usually be most effectively dealt with by generating thefeedback signal in response to data decisions which are relatively closein the output data decision stream to the data decision currently beingmade. Moreover, increasing the number of index pairs to encompass datadecisions which are more remote in time will have increasingly lesseffect in removing intersymbol interference, on the one hand, whilepossibly requiring increased hardware costs and/or processing time onthe other hand. In the present embodiment, the following (j₁,j₂) indexpairs are illustratively used: (1,1)(1,2)(1,3)(2,2)(2,3)(3,3). Themodulated weighted products of term D₁ (n) are thus given by ##EQU6##This term, then, encompasses a modulated weighted sum of all possibletwo-multiplicand products in which each multiplicand is one of the threeprevious data decisions.

The second term of signal D(n), D₂ (n), is similar to D₁ (n) except thatthe second multiplicand of each product is a complex conjugate andexcept that, per Eq. 4(a), the modulated harmonic is different. Anillustrative set of index pairs (j₃,j₄) for this second term is(1,1)(1,2)(1,3)(2,1)(2,2)(2,3)(3,1)(3,2)(3,3). Note that reversing theorder of the (j₃,j₄)index pairs, e.g., (1,2) and (2,1), providesdifferent products in term D₂ (n), although not in term D₁ (n). Thus,even though all of the indices j₃ and j₄ are each either 1, 2 or 3, justas in the case of indices j₁ and j₂, here there are nine possibledifferent decision/conjugate products, rather than six as in the case ofterm D₁ (n).

The final three terms of signal D(n), D₃ (n), D₄ (n) and D₅ (n), providecompensation for third-order intersymbol interference. (Again, in agiven application, it may be desired to use less than all of theseterms.) In particular, term D₃ (n) is comprised of a weighted sum ofthree-multiplicand products each having an associated weightingcoefficient and each modulating the appropriate harmonic from Eq. 4(a).Terms D₄ (n) and D₅ (n) are similar to term D₃ (n) but include one andtwo complex conjugates of data decisions, respectively. Index triples(k₁,k₂,k₃) for term D₃ (n) illustratively take on the values

(1,1,1)(1,1,2)(1,2,2)(1,2,3)(2,2,2,)(2,2,3)(3,2,3)(3,3,3). Index triplesfor terms D₄ (n) and D₅ (n) illustratively take on the values(1,1,1)(1,1,2)(1,2,1)(1,2,3)(1,3,2)(2,1,2)(2,2,1)(2,2,2)(2,2,3)(2,3,1)(2,3,2)(3,2,3)(3,3,2)(3,3,3)in each case. Note that for index triples (k₁,k₂,k₃), (k₄,k₅,k₆) and(k₇,k₈,k₉) less than all possible combinations yielding uniquethree-multiplicand decision/conjugate products are used. This is donesimply to minimize the amount of signal processing needed to generatesignal D(n). In general, using all possible combinations yielding uniquethree-multiplicand decision/conjugate products in generating terms D₃(n), D₄ (n) and D₅ (n) will provide additional reduction of third-orderintersymbol interference.

Compensation for fourth- or higher-order intersymbol interference may beprovided in accordance with the invention by obvious extension of thesecond- and third-order cases.

As previously indicated, the values of the weighting coefficients usedin feedback nonlinear processor 44, like those used in feedback linearprocessor 15, are adjusted adaptively in a decision-directed manner. Asin the case of processor 15, a gradient adaptation criterion isillustratively used. By way of example, this criterion is expressed forthe coefficients ##EQU7##

The other four sets of weighting coefficients used in generating termD(n) are generated similarly to coefficients ##EQU8## That is, ##EQU9##Although, in general, multiplicative scaling factors γ₁ (n)-γ₅ (n) canbe updated at each sampling time, they, like γ₀ (n), can more simply befractional constants, the values of which are determined empirically. Asseen in FIG. 1, feedback nonlinear processor 44, like feedback linearprocessor 15, receives signal E(n) for purposes of coefficient updating.

Feedback nonlinear signal D'(n) of FIG. 2 is similar to signal D(n)except that the multiplicative harmonics are given by Eq. 4(b). (Thesame index pairs and triples can be used, however.) Thus, D'(n)=##EQU10##

In addition, the coefficient updating relations for the FIG. 2embodiment are given by ##EQU11##

As previously discussed, all the Φ_(x),y 's illustratively have thevalue θ(n) in this embodiment which parameter could be provided, forexample, from carrier recovery circuit 24.

Attention is now directed to FIG. 3, which shows an illustrativeembodiment of feedback nonlinear processor 44. Feedback nonlinearprocessor 44' of FIG. 2 may be substantially the same except that,again, the harmonics which the decision/conjugate products modulate (inCSM units 71-75) are given by Eq. 4(b) rather than Eq. 4(a).

Processor 44 includes multiplexers 51-55, complex multipliers 61-65,coefficient store and multiplier (CSM) units 71-75 and accumulators81-85. During each sampling period, the serially connected chain ofmultiplexer 51, multiplier 61 and CSM unit 71 generates and stores thesix modulated weighted products of term D₁ (n) of signal D(n) inaccumulator 81. The modulated weighted products of terms D₂ (n)-D₅ (n)are generated and stored similarly, each by its ownmultiplexer-multiplier-CSM unit-accumulator chain. Since the chainswhich begin with multiplexers 51 and 52 generate terms of signal D(n)which have two-multiplicand decision/conjugate products, i.e., terms D₁(n) and D₂ (n), those multiplexers each extend two output leads totwo-input complex multipliers 61 and 62, respectively. Multiplexers53-55 each extend three output leads to multipliers 63-65, respectively,in order to generate the three-multiplicand decision/conjugate productswhich make up terms D₃ (n)-D₅ (n).

After terms D₁ (n)-D₅ (n) have all been stored in their respectiveaccumulators, they are added together in adder 86 to generate feedbacknonlinear signal D(n) on lead 45.

Processor 44 operates under the control of a clock 91. The latter, inturn, operates at a frequency sufficient to ensure that the generationof signal D(n) is completed during a single sampling interval, T. Asdescribed in further detail below, the clock pulses on output lead 91aof clock 91 control the shifting through processor 44 of serial bitstreams representing data decisions, complex conjugates of datadecisions and intermediate products of these with each other andharmonics of the carrier frequency. The clock pulses on lead 91a are, inaddition, received by a divide-by-twelve counter 93. The output pulsesfrom counter 93 on lead 93a initiate multiplication operations inmultipliers 61-65 and CSM units 71-75.

Counter 93 also drives selection register 92--illustratively adivide-by-seventeen counter. Register 92 increments the five-bit numberrepresented by the signals on its five output leads 92a by one count inresponse to each pulse from counter 93 on lead 93a. During each samplingperiod, three previous data decisions--A(n-1), A(n-2) and A(n-3)--arereceived by multiplexers 51-55 from delay line 15a of feedback linearprocessor 15, the two feedback processors advantageously sharing thisdelay line between them. Each of the three decisions is illustrativelyrepresented by twelve serial bits which are stored internally by eachmultiplexer in response to the first twelve clock pulses on lead 91a.The count on leads 92a at any given time indicates to each of themultiplexers which of the three decisions is to be provided on each ofthe multiplexer output leads in response to each group of twelve clockpulses.

By way of illustration, operation of the chain which begins withmultiplexer 51 in generating term D₁ (n) will now be described, theoperation of the other chains being similar. For purposes ofexplanation, it is assumed that the first sixty clock pulses within then^(th) sampling period have elapsed. Thus, at this point, the firstthree modulated weighted products of term D₁ (n) have been summed andstored in accumulator 81. The fourth modulated weighted product,##EQU12## has just been generated in CSM unit 71, while the fifth,unweighted modulated product, A(n-2)A(n-3)e^(j2)πf.sbsp.c^(nT) has justbeen generated in multiplier 61.

A number of operations occur concurrently in response to the next twelveclock pulses. The twelve bits representing ##EQU13## are shifted vialead 71a from CSM unit 71 into accumulator 81, where it is added to thecurrent contents of the accumulator. In addition, the unweighted, fifthproduct A(n-2)A(n-3)e^(j2)πf.sbsp.c^(nT) is shifted via lead 61a frommultiplier 61 into CSM unit 71. The binary count on leads 92a is now00101. In response to that count and to the twelve clock pulsescurrently being generated, multiplexer 51 provides the decision A(n-3)on both of its output leads 51a and 51b since the sixth (and last) valueof index pair (j₁,j₂) is (3,3). The subsequent pulse on lead 93ainitiates the accumulation operation in accumulator 81. It alsoinitiates the multiplication in CSM unit 71 of A(n-2)A(n-3) with thecomplex conjugate of the current value of its associated weightingcoefficient, ##EQU14## stored in the CSM unit. The pulse on lead 93aalso initiates the multiplication in multiplier 61 of decision A(n-3) byitself and by the harmonic e^(j2)πf.sbsp.c^(nT). The latter may beprovided in any of several ways, such as from the oscillator section ofcarrier recovery circuit 24. (For drawing clarity, a specific leadconnection from the latter is not shown in the drawing.)

The count on leads 92a, in addition to the function described above, isalso used to indicate to the various components of feedback nonlinearprocessor 44 when and when not to respond to the clock pulses on lead91a. For example, the fact that the last, i.e., sixth, two-multiplicanddecision/conjugate product in term D₁ (n) has now been generated ismanifested by the fact that the count on leads 92a is 00101. Simplelogic circuitry within multiplexer 51 and multiplier 61 precludes themfrom responding to further clock pulses. CSM unit 71 begins and ceasesoperation twelve clock pulses after multiplexer 51 and multiplier 61begin and cease their operation; for accumulator 81 the number istwenty-four clock pulses. Thus, similar logic circuitry in CSM units 71and accumulator 81 allows them to respond to clock pulses only when thecount on leads 92a is at or between 00001 and 00110, for the former, and00010 and 00111 for the latter. The other components within each chainof processor 44 similarly have logic circuits for controlling whichclock pulses they will respond to, depending on (a) how many productsare to be computed in that chain and (b) the position of the particularcomponent within its chain. A typical such logic circit is shown in theillustrative embodiment of CSM unit 71 in FIG. 4, as described below.

When the count on leads 92a has reached 01111, terms D₁ (n)-D₅ (n) haveall been generated and stored in accumulators 81-85, respectively. ANDgate 94 now generates a pulse on lead 94a which causes the contents ofaccumulators 81-85 to be added together in adder 86, the resultantsignal on lead 45 being feedback nonlinear signal D(n). When the counton leads 92a reaches its last value, 10000, the output of NOT gate 95 onlead 95a goes low, clearing multiplexers 51-55, multipliers 61-65, CSMunits 71-75, and accumulators 81-85 in preparation for generatingfeedback nonlinear signal D(n+1) during the next, (n+1)^(st), samplingperiod.

It will be appreciated that FIG. 3 represents but one of numerouspossible approaches for realizing feedback nonlinear processor 44 (or,as previously indicated, processor 44'). Thus, for example, the terms D₁(n)-D₅ (n) could be generated serially, one after the other, rather thanin parallel. Such an approach would require less arithmetic hardware.However, the circuitry needed to manipulate the decisions and complexconjugate and their products would be more complicated. In addition, allof the arithmetic operations which have to be performed in generatingsignal D(n) would still have to be completed during a single samplingperiod, imposing more stringent requirements on the speed with which thevarious arithmetic operations would have to be performed. Theserequirements might be advantageously satisfied by generating signal D(n)using a microprocessor. In any event, it will be appreciated that theneeds of the particular application will govern the structure ofprocessor 44.

Each of the functional blocks depicted in FIG. 3 may be of conventionaldesign and need not be described in further detail herein. However, aparticularly advantageous realization for CSM unit 71 (CSM units 72-75being similar) is shown in FIG. 4.

Each component of CSM unit 71 received clock pulses via lead 113a. Aspreviously indicated, CSM unit 71 is to operate only when the count onleads 92a is at or between 0001 and 0110. This mode of operation isachieved by logic circuit 113 which controls the flow of clock pulsesonto lead 113a from lead 91a in response to the count on leads 92a.

It will be remembered from Eq. (6) that the updated value of eachcoefficient is given by its previous value plus a term which includessignal E(n). The latter, however, is not known until signal D(n) hasbeen generated. Thus, as shown in FIG. 4, the modulated productsA(n-j₁)A(n-j₂)e^(j2)πf.sbsp.c^(nT) received by CSM unit 71 are delayedin a serial in/serial out shift register 101 such that as the firstmodulated product of term D₁ (n), A(n-1)² e^(j2)πf.sbsp.c^(nT), emergesfrom shift register 101 at the beginning of the next, (n+1)^(st),sampling period, signal E(n) is first becoming available on lead 22.A(n-1)² e^(j2)πf.sbsp.c^(nT) is multiplied in multiplier 103 by signalE(n) and by γ₁ (n)--illustratively a constant, γ₁ --to generate thecorrection term γ₁ A(n-1)² E(n)e^(j2)πf.sbsp.c^(nT).

At this time, coefficient ##EQU15## is just beginning to appear at theoutput of coefficient store 108, illustratively another serial in/serialout shift register. The correction term at the output of multiplier 103is added to ##EQU16## in adder 106 to generate ##EQU17## Since at thistime the first decision/conjugate product of term D₁ (n+1)--A((n+1)-1)²--is being introduced on lead 61a, coefficient ##EQU18## is passeddirectly to multiplier 110, so that the latter is able to form on lead71a the product ##EQU19## i.e., the first weighted modulated product ofterm D₁ (n+1). In addition, coefficient ##EQU20## is entered intocoefficient store 108 from which it will emerge for updating at thebeginning of the (n+2)^(nd) sampling period to generate ##EQU21##

Similarly, as each subsequent decision/conjugate product making up termD₁ (n+1) is introduced on lead 61a, the corresponding coefficientemerges from coefficient store 108, is updated, and is multiplied bythat product and the harmonic e^(j2)πf.sbsp.c^(nT) in multiplier 110. Atthe end of the (n+1)^(st) sampling period, the pulse on lead 95a clearssignal E(n) stored in multiplier 103 in preparation for storage thereinof signal E(n+1).

Althouh specific embodiments of my invention have been shown anddescribed, such merely illustrate the principles of my invention. Forexample, although the invention has been illustrated in conjunction witha QAM system, it is equally applicable to any modulated system in whichboth the carrier phase and amplitude are modulated, i.e.,information-bearing.

Thus, it will be appreciated that numerous arrangements embodying theprinciples of the invention may be devised by those skilled in the artwithout departing from their spirit and scope.

I claim:
 1. An arrangement for .[.equalizing.]. .Iadd.processing.Iaddend.samples of a received modulated data signal having apredetermined carrier frequency, said arrangement including meansoperative in response to said samples for forming decisions as to thevalues of data symbols represented thereby, each of said decisions beingrepresented by a complex number,characterized in that said arrangementfurther includes means (51-55, 61-65) for forming a plurality of signalproducts associated with an individual one of said samples, eachmultiplicand of each product being derived from a respective one of saiddecisions, each said respective one of said decisions bearing apredetermined temporal relationship to said one of said samples, atleast one multiplicand of individual ones of said signal products beingthe complex conjugate of the decision from which said one multiplicandis derived, and means (71-75) for multiplying each of .Iadd.at leastones of .Iaddend.said products by an associated coefficient and by apredetermined harmonic of said carrier frequency to form a plurality ofmodulated weighted products, .Iadd.and further characterized in that.Iaddend.said decision forming means .[.including.]. .Iadd.includes.Iaddend.data recovery means (14, 17) for forming a decision as to thevalue of the data symbol represented by said one of said samples inresponse to said modulated weighted products.
 2. The invention of claim1 wherein said modulated data signal is of the type in which both thecarrier phase and amplitude are information-bearing.
 3. The invention ofclaim 2 wherein said modulated data signal is a quadrature amplitudemodulated signal.
 4. The invention of claims 2 or 3 wherein saiddecision forming means includes means for applying to said data recoverymeans said modulated weighted products and a demodulated version of saidone of said samples.
 5. The invention of claim 4 wherein the harmonic bywhich each of .Iadd.said at least ones of .Iaddend.said products ismultiplied is of the form e^(j)[2πf.sbsp.c^(nT)(x-y-1)+Φ.sbsp.x,y^(]),where f_(c) is said carrier frequency, n is the sample time index, T isthe sample interval, x is the number of decisions in said each of saidproducts, y is the number of complex conjugates of decisions in saideach of said products and Φ_(x),y is a selected phase angle.
 6. Theinvention of claims 2 or 3 wherein said decision forming means furtherincludes means for combining said one of said samples with saidmodulated weighted products to form a combined modulated signal, meansfor demodulating said combined modulated signal and means for applyingthe demodulated signal to said data recovery means.
 7. The invention ofclaim 6 wherein the harmonic by which each of .Iadd.said at least onesof .Iaddend.said products is multiplied is of the forme^(j)[2πf.sbsp.c^(nT)(x-y)+Φ.sbsp.x,y^(]), where f_(c) is said carrierfrequency, n is the sample time index, T is the sample interval, x isthe number of decisions in said each of said products, y is the numberof complex conjugates of decisions in said each of said products andΦ_(x),y is a predetermined phase angle.
 8. An arrangement operativeduring each one of a plurality of successive sampling periods for.[.equalizing.]. .Iadd.processing .Iaddend.a respective one of asuccession of .Iadd.linearly equalized .Iaddend.samples of a receivedmodulated data signal having a predetermined carrier frequency, saidarrangement including feedback means (FIG. 1, 44; FIG. 2, 44') forgenerating an individual feedback signal associated with said onesample, and decision forming means (FIG. 1-12, 14, 17, 46; FIG. 2-12,14, 17, 46') jointly responsive to said one sample and its associatedfeedback signal for forming a decision as to the value of .[.the.]..Iadd.a .Iaddend.data symbol represented by said one sample, said onesample and said decision being represented by respective complexnumbers,characterized in that said feedback signal includes a pluralityof signal products .[.each.]. .Iadd.at least ones of which are.Iaddend.multiplied by an associated coefficient and by a predeterminedharmonic of said carrier frequency, each multiplicand of each signalproduct being derived from a respective decision formed by said decisionforming means during a previous one of said sampling periods, each saidrespective decision bearing a predetermined temporal relationship tosaid one sample, at least one multiplicand of individual ones of saidsignal products being the complex conjugate of the decision from whichsaid one multiplicand is derived.
 9. The invention of claim 8 whereinsaid modulated data signal is of the type in which both the carrierphase and amplitude are information-bearing.
 10. The invention of claim9 wherein said modulated data signal is a quadrature amplitude modulatedsignal.
 11. The invention of claims 9 or 10 wherein said decisionforming means includes means (14, 46) for combining said individualfeedback signal with a demodulated version of said one sample to form adata recovery input signal, and means (17) responsive to said datarecovery input signal for forming said decision.
 12. The invention ofclaim 11 wherein the harmonic by which each of .Iadd.said at least onesof .Iaddend.said products is multiplied is of the forme^(j)[2πf.sbsp.c^(nT)(x-y-1)+Φ.sbsp.x,y^(]), where f_(c) is said carrierfrequency, n is the sample time index, T is the sample interval, x isthe number of decisions in said each of said products, y is the numberof complex conjugates of decisions in said each of said products andΦ_(x),y is a selected phase angle.
 13. The invention of claims 9 or 10wherein said decision forming means includes means (46') for combiningsaid individual feedback signal with said one sample to form a combinedmodulated signal, means (12) for demodulating said combined modulatedsignal and means (17) responsive to the demodulated combined signal forforming said decision.
 14. The invention of claim 13 wherein theharmonic by which each of .Iadd.said at least ones of .Iaddend.saidproducts is multiplied is of the forme^(j)[2πf.sbsp.c^(nT)(x-y)+Φ.sbsp.x,y^(]), where f_(c) is said carrierfrequency, n is the sample time index, T is the sample interval, x isthe number of decisions in said each of said products, y is the numberof complex conjugates of decisions in said each of said products andΦ_(x),y is a selected phase angle.
 15. A method operative during eachone of a plurality of successive sampling periods for .[.equalizing.]..Iadd.processing .Iaddend.a respective one of a succession of complexsamples of a received modulated data signal having a predeterminedcarrier frequency, said method including the steps of generating anindividual feedback signal associated with said one sample, and forminga complex decision as to the value of the data symbol represented bysaid one sample in response to said one sample and its associatedfeedback signal,characterized in that said feedback signal includes aplurality of signal products .[.each.]. .Iadd.at least ones of which are.Iaddend.multiplied by an associated coefficient and by a predeterminedharmonic of said carrier frequency, each multiplicand of each signalproduct being derived from a respective decision formed during aprevious one of said sampling periods, each said respective decisionbearing a predetermined temporal relationship to said one sample and atleast one multiplicand of individual ones of said signal products beingthe complex conjugate of the decision from which said one multiplicandis derived.
 16. The invention of claim 15 wherein said modulated datasignal is of the type in which both the carrier phase and amplitude areinformation-bearing.
 17. The invention of claim 16 wherein saidmodulated data signal is a quadrature amplitude modulated signal. 18.The invention of claims 16 or 17 wherein said decision forming stepincludes the steps of combining said individual feedback signal with ademodulated version of said one sample to form a data recovery inputsignal, and forming said decision in response to said data recoveryinput signal.
 19. The invention of claim 18 wherein the harmonic bywhich each of .Iadd.said at least ones of .Iaddend.said products ismultiplied is of the form e^(j)[2πf.sbsp.c^(nT)(x-y-1)+Φ.sbsp.x,y^(]),where f_(c) is said carrier frequency, n is the sample time index, T isthe sample interval, x is the number of decisions in said each of saidproducts, y is the number of complex conjugates of decisions in saideach of said products and Φ_(x),y is a selected phase angle.
 20. Theinvention of claims 16 or 17 wherein said decision forming step includesthe steps of combining said individual feedback signal with said onesample to form a combined modulated signal, demodulating said combinedmodulated signal and forming said decision in response to thedemodulated combined signal.
 21. The invention of claim 20 wherein theharmonic by which each of .Iadd.said at least ones of .Iaddend.saidproducts is multiplied is of the forme^(j)[2πf.sbsp.c^(nT)(x-y)+Φ.sbsp.x,y^(]), where f_(c) is said carrierfrequency, n is the sample time index, T is the sample interval, x isthe number of decisions in said each of said products, y is the numberof complex conjugates of decisions in said each of said products andΦ_(x),y is a selected phase angle.
 22. A method for use in a receiver ofthe type in which a decision is formed during the n^(th) one of aplurality of successive periods of duration T, said decision beingformed by quantizing a respective demodulated sample of a receivedcomplex modulated data signal having a carrier frequency f_(c), both thecarrier amplitude and phase of said signal being information bearing,said decision being represented by a complex number A(n), said methodincluding the steps ofcombining a feedback signal V(n) with said sampleto form a signal Y(n) and forming said decision in response to saidsignal Y(n), said method characterized in that said signal V(n) includesthe terms ##EQU22## where * indicates complex conjugate and Φ₂,0 andΦ₁,1 are selected phase angles, the coefficients ##EQU23## havingrespective values associated with said decision A(n), and the indexpairs (j₁,j₂) and (j₃,j₄) having respective sets of predeterminedvalues.
 23. The invention of claim 22 comprising the further steps.Iadd.of .Iaddend.forming a signal E(n) in response to said signal Y(n)and said decision A(n), and updating the coefficients ##EQU24## inaccordance with ##EQU25## γ.sub. (n) and γ₂ (n) being selected scalingfactors.
 24. The invention of claim 23 wherein said signal V(n) furtherincludes the term ##EQU26## the coefficients ##EQU27## having respectivevalues associated with said decision A(n) and the index i having a setof predetermined values, and wherein said method includes the furthersteps ofupdating said coefficients ##EQU28## in accordance with##EQU29## γ.sub. (n) being a selected scaling factor.
 25. The inventionof claim 22 further characterized in that said signal V(n) furtherincludes at least a selected one of the terms ##EQU30## where Φ₃,0, Φ₂,1and Φ₁,2 are selected phase angles, the coefficients ##EQU31## havingrespective values associated with said decision A(n), and the indextriples (k₁,k₂,k₃), (k₄,k₅,k₆) and (k₇,k₈,k₉) having respective sets ofpredetermined values.
 26. The invention of claim 25 including thefurther steps offorming a signal E(n) in response to said signal Y(n)and said decision A(n), and updating the coefficients ##EQU32## inaccordance with ##EQU33## γ.sub. (n), γ₄ (n) and γ₅ (n) being selectedscaling factors.
 27. A method for use in a receiver of the type in whicha decision is formed during the n^(th) one of a plurality of successiveperiods of duration T, said decision being as to the value of arespective demodulated complex sample of a received data signalmodulated at a carrier frequency f_(c), both the carrier amplitude andphase of said signal being information bearing, said decision beingrepresented by a complex number A(n), said method including the stepsofcombining a feedback signal V(n) with said sample to form a signalY(n) and forming said decision in response to said signal Y(n), saidmethod characterized in that said signal V(n) includes the terms##EQU34## where * indicates complex conjugate and Φ₂,0 and Φ₁,1 areselected phase angles, the coefficients ##EQU35## having respectivevalues associated with said decision A(n), and the index pairs (j₁,j₂)and (j₃,j₄) having respective sets of predetermined values.
 28. Theinvention of claim 27 comprising the further stepsforming a signal E(n)in response to said signal Y(n) and said decision A(n), and updating thecoefficients ##EQU36## in accordance with ##EQU37## γ.sub. (n) and γ₂(n) being selected scaling factors.
 29. The invention of claim 28wherein said signal V(n) further includes the term ##EQU38## thecoefficients ##EQU39## having respective values associated with saiddecision A(n) and the index i having a set of predetermined .[.value.]..Iadd.levels.Iaddend., and wherein said method includes the furthersteps ofupdating said coefficients ##EQU40## in accordance with##EQU41## γ.sub. (n) being a selected scaling factor.
 30. The inventionof claim 27 further characterized in that said signal V(n) furtherincludes at least a selected one of the terms ##EQU42## where Φ₃,0, Φ₂,1and Φ₁,2 are selected phase angles, the coefficients ##EQU43## havingrespective values associated with said decision A(n), and the indextriples (k₁,k₂,k₃), (k₄,k₅,k₆) and (k₇,k₈,k₉) having respective sets ofpredetermined values.
 31. The invention of claim 30 including thefurther steps offorming a signal E(n) in response to said signal Y(n)and said decision A(n), and updating the coefficients ##EQU44## inaccordance with ##EQU45## γ.sub. (n), γ₄ (n) and γ₅ (n) being selectedscaling factors. .Iadd.
 32. Apparatus for processing a passband signalrepresenting a succession of complex signal values, said apparatuscomprising means for forming a succession of weighted sums of products,individual multiplicands of at least ones of said products being derivedfrom individual decisions as to respective ones of said signal values,at least ones of said individual multiplicands being the complexconjugates of the decisions from which they were derived and each of atleast ones of said products being multiplied by a respective harmonic ofthe carrier frequency of said passband signal, and means for forming atleast ones of said decisions in response to said passband signal andones of said weighted sums. .Iaddend..Iadd.
 33. The invention of claim32 wherein said decision forming means comprises means for generating asuccession of complex linearly equalized samples of said passbandsignal, and means for forming each of said ones of said decisions inresponse to at least a respective one of said equalized samples and atleast a respective one of said weighted sums. .Iaddend..Iadd.
 34. Theinvention of claim 33 wherein each of said individual decisions bears apredetermined temporal relationship to said respective one of saidsamples. .Iaddend..Iadd.
 35. The invention of claims 32, 33 or 34further comprising means for updating the coefficients used to form saidweighted sums in a decision-directed manner. .Iaddend. .Iadd.
 36. Amethod for processing a passband signal representing a succession ofcomplex signal values, said method comprising the steps of forming asuccession of weighted sums of products, individual multiplicands of atleast ones of said products being derived from individual decisions asto respective ones of said signal values, at least ones of saidindividual multiplicands being the complex conjugates of the decisionsfrom which they were derived and each of at least ones of said productsbeing multiplied by a respective harmonic of the carrier frequency ofsaid passband signal, and forming at least ones of said decisions inresponse to said passband signal and ones of said weighted sums..Iaddend..Iadd.
 37. The invention of claim 36 wherein said decisionforming step comprises the steps of generating a succession of complexlinearly equalized samples of said passband signal, and forming each ofsaid ones of said decisions in response to at least a respective one ofsaid equalized samples and at least a respective one of said weightedsums. .Iaddend..Iadd.
 38. The invention of claim 37 wherein each of saidindividual decisions bears a predetermined temporal relationship to saidrespective one of said samples. .Iaddend..Iadd.
 39. The invention ofclaims 36, 37 or 38 comprising the further step of updating thecoefficients used to form said weighted sums in a decision-directedmanner..Iaddend.